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Realising wreath products of cyclic groups as Galois groups. (English) Zbl 0662.12010
Let K be a field of characteristic zero, T, X indeterminates algebraically independent over K. For $$n\geq 1$$, let k(n)$$\geq 2$$ be an integer, $$f_ n(X,T)=X^{k(n)}+T$$. Put $$F_ 1(X,T)=f_ 1(X,T)$$ and define for $$n\geq 1$$, $$F_{n+1}(X,T)=F_ n(f_{n+1}(X,T),T).$$
In theorem 1 the author proves that if $$\overline{K}$$ is the algebraic closure of K then the Galois group of $$F_ n(X,T)$$ over $$\overline{K}(T)$$ is isomorphic to the wreath product $$\Gamma_ n=G_ 1[...[G_ n]...]$$ where for each $$i\leq n$$, $$G_ i$$ is the cyclic group of order k(i) with its natural permutation action on the symbols 1,...,k(i).
As a corollary the author proves that if K is a Hilbertian field containing the k(i)-th roots of 1 for $$i\leq n$$ then given $$t>1$$ there is a finite Galois extension L over K such that Gal(L/K) is isomorphic to the direct product of t copies of $$\Gamma_ n$$.
Reviewer: T.Soundararajan

##### MSC:
 11R32 Galois theory 12F10 Separable extensions, Galois theory 20F29 Representations of groups as automorphism groups of algebraic systems 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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